>> The other factor we need to consider is that the polynomial data complexity of Datalog is based on the most basic form, which doesn’t have function symbols (

>> The other factor we need to consider is that the polynomial data complexity of Datalog is based on the most basic form, which doesn’t have function symbols (ie datatypes like lists) or arithmetic. Once you start adding such features (which almost all real-world Datalogs do), the data complexity inevitably goes up. The informal definition of Datalog is that it is a decidable subset of Prolog, or a pure set of Prolog, etc, but I think it's more accurate to say that datalog clauses are a subset of the definite clauses such that a definite clause, C, is datalog if and only if a) C has no functions other than constants and b) every variable in the head literal of C is shared with a literal in the body of C. For example, below, C is datalog, D and E are not: C = P(x,y) ← Q(x,z), R(z,y) % datalog D = P(x,y) ← Q(x,z) % z is not shared with head(D) E = P(x,y) ← Q(f(x),z), R(z,y) % f(x) is not a constant function. Note that Prolog programs are sets of definite clauses, but without the restrictions (a) and (b) of datalog. As the article points out, lists need functions, so for example p([H|T], H) is not datalog because [H|T] is a function, and for arithmetic n(s(0)) is not datalog because s(0) is a function. So what's up with functions? In logic programming a function is a function symbol followed by n terms, where n is the arity of the function symbol, so if f(x,y) is a function then f is a function symbol with arity 2. Unlike in imperative programming, functions in logic programming _are not replaced by their values_, instead the truth of a formula containing functions is calculated by substituting the variables in the function. So f(x) can become f(a), f(b) etc. where a and b are constants. Problem is, variables in logic programming can be substituted for arbitrary terms and functions themselves are terms, so given the function symbol f of arity 1 we can make an infinite number of functions: f(x), f(f(x)), f(f(f(x))), f(f(f(f(x)))), etc. That's one important reason for the undecidability of First Order Logic (FOL). Definite logic, the logic of definite clauses, which is a restriction of FOL, is _semi_ decidable, meaning that if a theorem doesn't have a proof, there is no way to know in finite time. Datalog doesn't have functions so it removes this particular source of undecidability. If, in addition to not having functions, the Herbrand Universe (the set of all constants) is finite, then datalog is decidable, informally because every proof can only replace the variables in a program with a finite set of constants, which means there's a finite set of proofs for every program. What about the requirement that the head literal shares variables with the body literals? That's to ensure that the entire datalog program can be ground. Datalog is evaluated "bottom up" meaning that literals in the body are ground before literals in the head, so in a clause like D above, even if Q(x,z) is grounded during evaluation, the variable y in the head of the clause cannot be grounded because it's not shared with the body. Which means that we can't prove that P(x,y) is a consequence of Q(x,z). In a clause like C, on the other hand, we can ground every variable by grounding the literals in the body and so bottom-up evaluation is possible. Bottom-up evaluation is desired because it avoids left-recursions and it makes the ordering of clauses irrelevant for the proof of a program, unlike in Prolog. This is an attractive property of datalog. Now, about functions, the Inductive Logic Programming community knows a few tricks to retain the decidability of datalog programs even with functions. The sneakiest such trick is "flattening" where functions are "hidden" away from the datalog clauses. For instance, instead of writing: member(X, [X|Xs]) ← ... With flattening we can write: member(X, Xs) ← head(Xs,X), ... Where "head" is defined as head([X|Xs],X). With flattening we can keep some parts of the program datalog and the rest definite, and so have our cake and eat it.